CS 581 Tandy Warnow Todays material from Chapters

Today’s material (from Chapters 1 -3) • Newick strings • Representing rooted trees using

Newick representations • For a rooted tree, we represent a graph with a string

Rooted vs. unrooted • Task: be able to move between rooted and unrooted representations

Clades Definition: Let T be a rooted tree leaf-labelled by S, let v an

Triplet Trees Definition: Let T be a rooted tree leaf-labelled by S. A triplet

Computing rooted trees from clades • Partially order the set of clades by containment,

Tree construction from clades Questions: • Accuracy? • Running time? • But, how are

Clade compatibility • Definition: Let T be a rooted tree leaf-labelled by S, v

Proof of theorem • One direction is easy • The other direction is a

Rooted Tree Compatibility • Input: Set X of rooted trees, not all on the

ASSU algorithm Given set X of k triplet trees on n species: • If

Why does it work? If the set X of triplet trees is compatible, •

Compatibility of rooted trees • Suppose the input is a set X of rooted

Summary (so far) • We have seen how to construct a rooted tree from

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CS 581 Tandy Warnow

Today’s material (from Chapters 1 -3) • Newick strings • Representing rooted trees using clades and rooted triplet trees • Constructing a rooted tree from its set of clades using Hasse Diagrams • Constructing a rooted tree from rooted triplet trees using Aho, Sagiv, Szymanski, and Ullman • Constructing a rooted tree from rooted subtrees of any size

Newick representations • For a rooted tree, we represent a graph with a string with the taxa, commas, and nested parentheses. • For example, what tree is represented by (a, (b, (c, ((d, e), (f, g))))))? • How do we represent an unrooted tree? (Easy - root it somewhere, and write down the Newick representation of the rooted version. )

Rooted vs. unrooted • Task: be able to move between rooted and unrooted representations of trees • Task: be able to compare two trees and see if they are different or the same

Clades Definition: Let T be a rooted tree leaf-labelled by S, let v an internal node in T, and let Xv be the set of leaves in T below v. Let Clades(T) = {Xv: v in V(T)}. Note: Xv is also called the “cluster” at node v, so this is sometimes called Clusters(T). • Question: Given Clades(T), can we compute T?

Triplet Trees Definition: Let T be a rooted tree leaf-labelled by S. A triplet tree is a rooted 3 -leaf subtree of T, such as ((a, b), c). The set of all triplet trees of T is denoted Triplets(T). • Question: Given Triplets(T), can we compute T?

Computing rooted trees from clades • Partially order the set of clades by containment, add in the full set S, and compute the Hasse Diagram of the resultant poset (partially ordered subset) Note: Hasse Diagrams and Partially Ordered Sets are explained in Appendix B in the textbook.

Tree construction from clades Questions: • Accuracy? • Running time? • But, how are we to compute clades?

Clade compatibility • Definition: Let T be a rooted tree leaf-labelled by S, v an internal node in T, and Xv the leaves in T below v. Let Clades(T)={Xv: v in V(T)}. • Theorem: Let X be a set of subsets of S. Then there exists a tree T such that X = Clades(T) if and only if for all A, B in X, either A and B are disjoint, or one contains the other.

Proof of theorem • One direction is easy • The other direction is a proof by construction!

Rooted Tree Compatibility • Input: Set X of rooted trees, not all on the same set of leaves. • Output: Tree T (if it exists) that agrees with all the trees in X, and otherwise “Fail” This problem is solvable in polynomial time. Proof: the Aho, Sagiv, Szymanski, and Ullman (ASSU) algorithm!

ASSU algorithm Given set X of k triplet trees on n species: • If n>1, then construct graph with each species one of the vertices, and edges (a, b) for triplets ab|c. • If the graph has a single component, reject (the set is not compatible); else recurse on each component, and return tree formed by making the rooted trees on the components each a subtree off the root of the returned tree.

Why does it work? If the set X of triplet trees is compatible, • Then there is a rooted tree T with at least two subtrees off the root, T 1 and T 2. • Any two leaves a, b in the same subtree cannot be in a triplet ab|c. • Hence the graph formed for the set of triplet trees cannot be connected. • Therefore the graph formed for the set of triplet trees must have at least two components. • This argument applies recursively to every subset of X. • Hence the algorithm returns a tree on which all the triplet trees agree. If the set X of triplet trees is not compatible, it is not hard to show that the algorithm will detect this (proof by induction on the number of taxa).

Compatibility of rooted trees • Suppose the input is a set X of rooted trees (not necessarily triplet trees). • Can we use ASSU to determine if X is compatible, and to compute a compatibility supertree for X? • Solution: YES, just encode each rooted tree in X by its set of rooted triplet trees (or some subset of these that suffices to define each tree in X), and then run ASSU.

Summary (so far) • We have seen how to construct a rooted tree from its set of clades or triplet trees. • We have seen how to test compatibility of a set of clades or rooted trees. How do we use these techniques for constructing unrooted trees?

Summary (so far) • We have seen how to construct a rooted tree from its set of clades or triplet trees. • We have seen how to test compatibility of a set of clades or rooted trees. Can we use these ideas to design divide-andconquer methods to construct large rooted trees?

Summary (so far) • We have seen how to construct a rooted tree from its set of clades or triplet trees. • We have seen how to test compatibility of a set of clades or rooted trees. What can we do to construct unrooted trees?